Premium
On the Points on the Unit Circle with Finite b –Adic Expansions
Author(s) -
Schatte Peter
Publication year - 2000
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/1522-2616(200006)214:1<105::aid-mana105>3.0.co;2-k
Subject(s) - mathematics , unit circle , unit (ring theory) , integer (computer science) , combinatorics , group (periodic table) , base (topology) , prime (order theory) , sequence (biology) , mathematical proof , finite set , set (abstract data type) , geometry , mathematical analysis , chemistry , mathematics education , organic chemistry , biology , computer science , genetics , programming language
Let $b > 1$ be an arbitrary integer base and let $l \ge 0$ be the number of different prime factors $p_j$ of $b$ with $p_j \equiv 1 \, {\rm mod} \, 4$ , $j = 1, \ldots , l$ . Further let $\Pi _b$ be the set of points on the unit circle with finite $b$ –adic expansions of their coordinates and let $\Phi _b$ be the set of angles of the points $P \epsilon \Pi _b$ . Then $\Phi _b$ is an additive group which is the direct sum of $l$ infinite cyclic groups and of the finite cyclic group $\{ \pi / 2 \}$ . If in case of $l > 0$ the points of $\Pi _b$ are arranged according to the number of digits of their coordinates, then the arising sequence $P_0, \, P_1, \ldots$ is uniformly distributed on the unit circle. On the other hand, in case of $l = 0$ the only points in $\Pi _b$ are the exceptional points (1, 0), (0, 1), (–1, 0), (0, –1). The proofs are based on a canonical form for all integer solutions $x, \, y$ of $x^2 + y^2 = b^{2k}$ .